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Bone Adaptation as a Geometric FlowBryce A. Besler, Tannis D. Kemp, Nils D. Forkert, Steven K. Boyd

Abstract—This paper presents bone adaptation as a geometricflow. The proposed method is based on two assumptions: first,that the bone surface is smooth (not fractal) permitting thedefinition of a tangent plane and, second, that the interfacebetween marrow and bone tissue phases is orientable. Thispermits the analysis of bone adaptation using the well-developedmathematics of geometric flows and the numerical techniques ofthe level set method. Most importantly, topological changes suchas holes forming in plates and rods disconnecting can be treatedformally and simulated naturally. First, the relationship betweenbiological theories of bone adaptation and the mathematicalobject describing geometric flow is described. This is termedthe adaptation function, F , and is the multi-scale link describedby Frost’s Utah paradigm between cellular dynamics and bonestructure. Second, a model of age-related bone loss termedcurvature-based bone adaptation is presented. Using previousliterature, it is shown that curvature-based bone adaptation isthe limiting continuous equation of simulated bone atrophy,a discrete model of bone aging. Interestingly, the parametersof the model can be defined in such a way that the flow isvolume-preserving. This implies that bone health can in principlechange in ways that fundamentally cannot be measured byareal or volumetric bone mineral density, requiring structure-level imaging. Third, a numerical method is described and twoin silico experiments are performed demonstrating the non-volume-preserving and volume-preserving cases. Taken together,recognition of bone adaptation as a geometric flow permits therecruitment of mathematical and numerical developments overthe last 50 years to understanding and describing the complexsurface of bone.

Index Terms—Bone Adaptation, Geometric Flow, Osteoporosis,Aging

I. INTRODUCTION

The concept of bone adaptation as a geometric flow ispresented. The work presented here is a significant exten-sion of a conference proceeding [1]. Particularly, an artifactof signed distance transforms of sampled signals has beenidentified [2] and solved in the case of computed tomographyimages of biphasic materials [3]. The mathematics have beenexpanded significantly to tightly link the model to the theoryof geometric flows.

II. ADAPTATION AS A GEOMETRIC FLOW

It is assumed that bone changes occur at the interface ofmarrow and bone tissue. As a consequence of this claim,with an assumption of smoothness, many statements can bemade about the underlying dynamics. Specifically, they can be

This work was supported by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada, grant RGPIN-2019-4135.

B.A. Besler and T. D. Kemp are in the McCaig Institute for Bone and JointHealth, University of Calgary Canada. N.D. Forkert is with the Department ofRadiology and the Hotchkiss Brain Institute, University of Calgary, Canada.S.K. Boyd is with the Department of Radiology and McCaig Institute for Boneand Joint Health, University of Calgary Canada e-mail: skboyd@ucalgary.ca

Manuscript received November 18, 2020; revised TODO

modeled as a geometric flow where the flow rate has a historicand important meaning in the theory of bone adaptation.

A. Biology of Bone Adaptation

Bone adaptation occurs fundamentally at the surface [4],[5], [6]. This is in opposition to ontogenesis [7] and indirectfracture healing [8] where endochondral ossification is re-placing cartilage or intramembranous ossification is occurringdirectly from sheets of mesenchymal connective tissue. Func-tional adaptation refers to adaptation controlled principallythrough mechanics, typically coordinated by the osteocytesummarizable by a biological set point theory termed themechanostat [5]. Adaptation is separated into modeling (mo-tion of the periosteal and endosteal surfaces through surfacedrifts) and remodeling (changes in cortical and trabecular bonethrough coordinated cell action) [5]. The unit of remodeling isthe basic multicellular unit (BMU) consisting of osteoblasts,osteoclasts, osteocytes, and other cells coordinated throughcellular dynamics. Remodeling is coordinated differently inthe lacunae of trabecular bone [9] and the osteon’s of corticalbone [10]. Furthermore, a distinction is made between changesin shape (external remodeling) and changes in the materialproperties (internal remodeling) [11]. The principle concept isthat adaptation occurs on the surface, which presupposed theexistence of a surface as opposed to a density field and thatadaptation can be modeled as a change in this surface overtime.

B. The Bone Surface

Let bone be described by a density field ρ : Ω→ R+ definedon a domain Ω ⊂ R3. It is assumed that bone is a biphasicmaterial consisting of the marrow phase and the bone tissuephase, the domain is a union of the two phases Ω = ΩMarrow∪ΩTissue where ΩMarrow and ΩTissue are the marrow and bonetissue components in the field, respectively. Importantly, sincethe bone is a biphasic material, its interface can be describedas an orientable surface:

C : R2 → R3 (1)

where C is the surface. The consequence of having an ori-entable surface is that there is a defined inside and outside, sothat a volume can be defined and area elements oriented.

One additional claim is made that the surface is locallysmooth, permitting differentiation. Since the surface is dif-ferentiable, the area is finite, a tangent plane can be de-fined at each point on the surface, and principle, mean, andGaussian curvature defined. This constraint will be relaxed inSection II-C to permit topological changes during adaptation.

In contrast to differential geometry [12], one could de-fine bone using the theory of fractal geometry [13]. Fractal

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(a) A Rod Resorbing (b) A Plate Forming a Hole (c) Periosteal Drift

Fig. 1. Idealized surface changes in bone structure. (1a) Rods can resorb, changing topology; (1b) plates can form holes, causing rod-to-plate transition; and(1c) periosteal drift can change the gross morphometry.

dimension is a well established morphometric parameter oftrabecular bone [14], [15] defined from fractal geometry andthere are strong arguments that bone has fractal properties [16].The coastline paradox is the quintessential natural experimentabout the origin of fractals, where the measured length of acoastline depends on the size of the ruler you measure it with.The paradox is that the area continues to increase as the rulerdecreases, all the way down to the scale of an atom, leading toone having a curve with finite volume but infinite area. In bone,the perimeter is replaced with surface area and the ruler withthe resolution of the imager. While this behavior is confirmedat in vivo resolutions [15], it is not clear if this trend wouldcontinue ad infinitum. At some resolution — say, the 100 nmscale — the measured area is assumed to stabilize. Scanningelectron microscopy confirms relatively smooth surfaces, albeitsome surface roughness [17]. While this theory holds thatthe surface is smooth, no claims are made on properties ofthe surface not exhibiting fractal-like behavior in the form ofpower laws, such as the distribution of pore size [18].

C. Adaptation as a Geometric Flow

Having established the bone surface as an orientable,smooth surface, attention is placed on how to move the surfacein time. This leads directly to an extrinsic geometric flow:

Ct = FN (2)

where ·t is a partial derivative in time, F is a rate of surfacegrowth that varies spatially across the surface, and N is thenormal of the surface. This equation captures motion only inthe normal direction along the surface since tangential motionleads to a reparameterization of the surface and not a changein its geometry. With an initial surface, this presents boneadaptation as an initial value (Cauchy) problem: Ct = FN (3a)

C(x, 0) = C0 (3b)

where C0 is the initial bone surface. Such a formulation hasbeen used extensively in computational physics [19] and activecontours [20].

Classic results of extrinsic geometric flows follow naturallyfrom presentation of the Cauchy problem [21], [22], [23].Changes in topology can occur as rods disconnect or holesform in plates. In finite time, the curve can develop sharpcorners, which are continuous but not smooth, requiring spe-cial treatment through the theory of viscous solutions [24].

A complete contrast and comparison of geometric flows andtheir relation to bone adaptation is beyond the scope of thiswork and is an area of future interest.

D. The Adaptation Function, F

The principle consequence of considering bone as a geo-metric flow is that the quantity F , combined with the initialsurface, completely describes the adaptation of bone. Due toits importance, we term the quantity F the adaptation function.

1) F in Frost’s mechanostat: F is exactly the mechanostatgraph of Carter [25] and Frost [26]. Remodeling, lamellar bonedrifts, and woven bone drifts are summarized in a single graphwhere changes in bone density (or surface) are a function oflocal mechanical strain. It is a summary of the BMU andchanges spatially across the surface of bone. This paradigmhas been used extensively to develop computational models ofbone adaptation [27], [28], [29], [30], [31]. Attempts have beenmade to measure F experimentally and establish the presenceof lazy zones [32].

2) F in Dynamical Systems: One paradigm for under-standing skeletal health is to treat the basic multicellularunit (BMU) as a dynamical system. Hormones circulating inthe blood stream (PTH, calcitonin, calcitriol, estrogens, etc.)and cytokines expressed locally (OPG, RANK, Wnts, TGF-β, etc.) control the rate of bone formation and resorption.Importantly, these substrates change the temporal dynamics ofother substances, say through the down regulation of PTH ascalcium leaves the bone and enters the blood stream or throughthe modified expression of RANK-L from osteoblasts, whichleads to nonlinear effects in neighboring cells. Dynamicalsystems based on nonlinear partial differential equations havebeen used extensively to model coupled system. At the cellularlevel, dynamical models have explained paradoxes in experi-mental research [33] as well as predicting the response of boneto different cytokines [34], [35]. The adaptation function, F ,is a continuous summary of the activity of discrete osteoblastsand osteoclasts, similar in concept to diffusion being a contin-uous summary of quantized particles moving from Brownianmotion. In this paradigm, the adaptation function is the linkbetween cellular- and tissue-scale dynamics [36], [37].

3) F in Cellular Automata: Dynamical systems lead im-mediately to the paradigm of cellular automata (although notequivalent, taken here in spirit with complex adaptive sys-tems and agent based modeling) [38], [39], [40]. The centralconcept of cellular automata is that complex behaviors can

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emerge from iterating simple rules, emerging in a way non-obvious by studying the rules in isolation. Such models havebeen used for predicting cortical remodeling [41], posing boneadaptation as a topological optimization problem [42], andsimulating fracture healing [43]. Viewed as agents, osteoblastand osteoclast cells can be thought to interact through simplerules, being defined by cytokines, local loading, and genetics.The system adapts through time, causing the spatial patternof bone to emerge. In contrast to dynamical systems, this is acomputational paradigm of bone adaptation. As in dynamicalsystems, the adaptation function F is a summary of these localagents.

E. Remarks on Internal and External Models

There is a rich history in the development of models ofbone adaptation. We differentiate those models that make thetwo-phase assumption from those that do not with Carter’sterminology of internal remodeling [44], [29] and externalremodeling [45], [27], [46], respectively. There are modelsthat make use of both [47]. Equation 2 is the central equationfor external remodeling while the central equation for internalremodeling is:

ρt = F (4)

where ρ is the density field.In both cases, the adaptation function F remains

the central quantity of investigation. In internal remod-eling methods, F has dimensions of density per unittime ([Mass][Length]-3[Time]-1) while in external remodel-ing methods, F has dimensions of length per unit time([Length][Time]-1). While these models are tightly coupledthrough F , their assumption on the density field have differentconsequences. For instance, internal remodeling can have ahole form in the center of a trabecula, not connected to themarrow space. Unless one relaxes the reinitialization condition(Section IV-A4), this cannot occur in external remodeling. Aconsequence of this is that the analytic models of Weinans [48]and Cowin [49] are not equivalent.

III. CURVATURE-BASED BONE ADAPTATION

We now present a specific model of bone adaptation forage-related bone loss. The model is based heavily on a priormodel of age-related bone loss called simulated bone atrophy(SIBA) [50], [51], [52], [53]. Using prior literature, it willbe demonstrated that SIBA simulates mean curvature flow,providing a strong link to this geometric model.

A. The Model

Curvature-based bone adaptation models age-relatedchanges in bone loss as a summation of advection of thesurface and mean curvature flow. This gives the adaptationfunction:

F = a− bκ (5)

where a is the advection constant with dimensions[Length][Time]-1, b is the curvature constant with dimensions[Length]2[Time]-1, and κ is the mean curvature with dimen-sions [Length]-1. While a can take on any value, b can only

take on positive values. Negative values of b are consistentwith inverse mean curvature flow, which is not defined forflat surfaces and becomes numerically unstable for non-flatsurfaces. This is a two-parameter model defining a spatiallyvarying adaptation function that depends only on the localgeometry.

The intuition behind Equation 5 is the same as in SIBA.Thin connections in the bone will resorb first not only becausethey are smaller, but because they have much higher curvature.Holes can form in plates causing plate-to-rod transition. Gener-ally, changes occur on the surface, the trabecular bone surfaceerodes, and the rate at which bone changes varies across thesurface.

An important definition is when F = 0 across the surface,as the bone will stop adapting. Rearranging Equation 5, thestopping condition can be found:

F = b(〈κ〉 − κ) (6)

where 〈κ〉 = a/b is the average mean curvature across thesurface. The bone will stop adapting when it is a surfaceof constant mean curvature, κ = 〈κ〉 everywhere. Such anequation is seen in the Young-Laplace equation [54], [55]describing soap films, surface tension, and capillary rise.

A closed form solution for curvature-based bone adaptationis difficult even for simple analytic surfaces. A solution isgiven for the sphere [56] and has been long known for thecases when a = 0 or b = 0. However, simple analytic surfacesare in someway unfaithful for developing intuitions on theequation.

B. Relation to Minimal Surfaces

A minimal surface is a surface of smallest area given aconstraint. Equivalently, as mean curvature is the first variationof area, a minimal surface will have a mean curvature ofzero everywhere. The first of such surfaces were Euler’scatenoid and helicod. Later, Schwarz and Neovius describedperiodic minimal surfaces that extended infinitely. Schoenlater classified these surfaces and discovered the gyroid [57]Finally, surfaces of non-zero and spatially varying constantmean curvature were explored by Chopp and Sethian [58],[59] using level set methods. Such surfaces have been used fordesigning scaffolds for tissue engineering [60] and lightweightbut strong structures [61].

C. Relation to SIBA

The relationship between curvature-based bone adaptationand simulated bone atrophy (SIBA) is now described. SIBAworks on binary images of bone where the bone tissue phase isthe foreground object and the marrow phase is the backgroundobject. A finite support Gaussian filtration is used to blur theobject followed by a threshold to rebinarize the image. Physi-cal interpretations were given to the variance of the Gaussianblur and threshold value based on osteoblast efficiency andresorption depth. The key methodological novelty of SIBAwas that it naturally handled bone changing topology, whererods could resorb and plats could form holes.

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The intuitive similarities between SIBA and curvature-basedbone adaptation is that Gaussian blurring moves the surface ina way that resembles mean curvature flow. Local changes onthe bone surface are dependent on the magnitude and sign ofthe local mean curvature. As such, an advection term is usedto model thresholding and a mean curvature term is used tomodel the Gaussian blurring. However, this link can be madeconcrete, and is done so now. The main result is that if athreshold of 0.5 is used in SIBA, the method is equivalent tomean curvature flow in the limit as the product of kernel sizeand epoch time goes to zero.

1) Advection — Threshold Link: The link between advec-tion and thresholding is derived following a previous derivationfor Gaussian smoothed surfaces (Appendix of [62]). Considera binary image, I , blurred with some Gaussian filter, Gσ:

J = I ∗Gσ (7)

where ∗ is convolution and J is the resulting grayscale image.We seek to understand how far the surface moves given athreshold T of J . The intensity is normalized by forcing thebinary image I to take on values of 0 or 1.

The curvature-dependence of the problem is removed byconsidering a blurring much smaller than the local meancurvature (σ |H|) such that the surface is near flat (relativeto the smoothing). Then, we can reduce the three-dimensionalproblem to a one-dimensional problem by considering the linealong the surface normal, since blurring does not change thesurface in the tangent plane. We define I to be a unit step(Heaviside) function θ(x) with zero crossing at x = 0. Theunit step is substituted into Equation 7 and the location of theT level set is computed.

J(x) = θ(x) ∗Gσ (8)

T =1

2

(1 + erf

(x√2σ

))(9)

x =√

2σerf−1 (2T − 1) (10)

where erf(·) is the error function:

erf(z) =2√π

∫ z

0

e−t2

dt (11)

and erf−1 the inverse error function. Note that the inverse errorfunction is unique if |z| < 1, which corresponds to thresholdsinside the dynamic range [0, 1] specified for the binary image.Furthermore, erf−1(0) = 0, corresponding to no change in thesurface if T = 0.5.

Equation 10 gives an estimation of the distance the surfacetravels in flat areas for a given threshold T . Dividing thisdistance by the sample time, which is the inverse of activationfrequency (AF) in SIBA, gives an estimate of the correspond-ing advection constant:

a =√

2σerf−1 (2T − 1) AF (12)

2) Mean Curvature — Gaussian Filtration Link: The rela-tionship follows in two steps. First, consider the heat flow ofthe image:

Iτ = α∆I on Ω× (0,∞) (13a)I(x, 0) = I0 on Ω× 0 (13b)

where I0 is the original binary image, τ ∈ [0,∞) is a time-like parameter, α is thermal diffusivity typically set to unity,and ∆ = ∇2 is the Laplacian operator. This defines a one-parameter family of images where the time-like parametercaptures the scale of objects in the image [63], [64]. It iswell known that the solution of the heat equation is Gaussianconvolution:

I(x, τ) = I0(x) ∗Gτ (x) (14)

where 2ατ = σ2 in Equation 7. As such, one can work withI(x, τ) equivalently to the Gaussian blurring. The problemnow reduces to comparing heat flow of the image to meancurvature flow of the surface.

Realizing the link between Gaussian convolution and heatflow, SIBA is exactly the same as the BMO (Bence-Merriman-Osher) algorithm in computational physics for simulatingmean curvature flow, except a threshold different from 0.5 isselected [65]. BMO simulates mean curvature flow by blurringa binary image using the heat equation and rebinarizing thefield with a threshold at 0.5. Evans (Theorem 5.1, [66])proved that if u is the viscous solution from mean curvatureflow and I(x, τ) the solution from the diffusion equation,the two methods are equivalent in the limit of small τ .The consequence is that for small values of σ/AF and withT = 0.5, SIBA simulates mean curvature flow.

Following the methods of the BMO algorithm [65] inspherical coordinates, the mean curvature constant can beestimated as twice the thermal diffusivity constant:

b = 2α (15)

Note that b and α have the same dimensions,[Length]2[Time]-1. Substituting into the relationship betweenα and σ:

b =σ2

τ(16)

where τ can be estimated as the inverse of activation fre-quency, as in the case of advection. The factors-of-two cancelbecause the surface is embedded in three dimensions. Finally,one can estimate the target mean curvature in SIBA by dividingEquation 12 by Equation 16:

〈κ〉 =

√2erf−1 (2T − 1)

σ(17)

Noting the restriction in Section III-C4, the algorithm will stopbefore this condition is met.

3) Limitations on these Similarities: It should be noted thatthe derivations in Section III-C1 and III-C2 are approximateand not exact. Additional analysis is needed to derive boundsand convergence orders. Differences will arise because SIBAis compositional (Gaussin blur then thresholding) while theproposed model is additive (summing advection and mean cur-vature flow). Finally, SIBA was designed to be implementedin discrete space with a finite support Gaussian filter, causingdifferences to this continuous model.

4) Advantages and Disadvantages: There are three primaryadvantages to curvature-based bone adaptation over simulatedbone atrophy. The first advantage is a high-order representationof the bone surface. Since the surface is represented as a

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signed distance transform, gradients are available, and sub-voxel shifts can be tracked over time. In SIBA, the bone isrepresented by a binary image that is continually blurred andrebinarized. This inherently limits the representation to firstorder accurate, O(h), since information on the derivatives islost by binarization. Furthermore, the bone surface can nevertraverse through voxel edges in very flat surfaces before beingrebinarized, causing the surface to stop advecting artificiallyearly (see Section 4 of [65]). This problem has an intricatelink to anti-aliasing filters in computer graphics [67], [2]. Thesecond advantage is well-defined mathematics for geometricflows. This gives us principled methods of understanding whenthe flow stops, if the flow minimizes an energy functional,and how the area and volume change with the flow. Furtherrefinement of the link between dynamic histomorphometry [4],[68], [69] and geometric flows [70] as well as formulatingenergy functionals for bone adaptation [71] is an excitingfuture direction. The final advantage is that building loaddriven adaptation models based on the binary representationhas the distinct disadvantage of requiring blurring to establishnormal vectors [46]. The Gaussian filtration has an inherenttrade-off where large blurs are needed to prevent quantizationof the normal vector while small blurs are preferred to limitstructural changes. Additionally, this causes an implicit meancurvature flow on top of the expected load driven adaptation,presenting difficulties in model validation.

The two disadvantages of the proposed method are that itrequires more memory and the algorithms are more difficultto implement. The memory requirement comes from needingto store the signed distance transform in a floating pointrepresentation where binary images can be stored with anunsigned 8-bit integer and massively compressed. Second, theproposed method requires specialized techniques for embed-ding and evolving, which are not yet standard in many imageprocessing libraries. The tools required to implement SIBAexist in virtually all image processing libraries.

Beyond contrasting, the two methods share a definingsimilarity: the ability to handle topological changes in boneduring adaptation. This is the defining feature of any externalremodeling algorithm and follows simply as a corollary ofthe assumption that bone adapts at the surface. If it adaptsat the surface, topological changes will occur, and they mustbe treated appropriately. Any algorithm that cannot handletopological changes, such as those assuming diffeomorphisms,are not appropriate for modeling adaptation. This explainswhy successful methods in the field of brain imaging [72]and shape analysis [73] have had limited success in modelingbone adaptation. These methods make the explicit assumptionof spatial normalization: that there exists a diffeomorphismbetween time points. While this appears true for brains, thisis not true of bone microarchitecture within the same subjector between subjects. Similarly, adaptation simulation methodsthat used deformations of the grayscale data have an implicitweak form of the topology assumption that only holds forshort time durations [74].

IV. NUMERICAL SIMULATION

A. Level Set Method

The level set method is used to simulate the geometricflow [75], [76], [77]. Primarily, the level set method representsthe curve implicitly as the zero level set of an embeddingfunction, φ:

C = x | φ(x) = 0 (18)

This prevents many issues seen in parametric representationsof surfaces [20], described in Figure 2. Importantly, an equa-tion of motion can be computed for the implicit surface basedon the curve evolution equation (Equation 2) and taking thetemporal derivative of the implicit contour (Equation 18):

φt + F |∇φ| = 0 (19)

Using this derivation, the equivalent initial value (Cauchy)problem can be stated using the implicit embedding function:

φt + F |∇φ| = 0 (20a)φ(x, 0) = ±d(x,C) (20b)

Instead of working with the curve directly, motion and mor-phometrics will be performed on the embedding, being ableto recover the curve as the zero level set of the embedding atanother time.

The finite difference method will be used to solve Equa-tion 20a numerically. In general, the finite volume methodis inappropriate for this solver as bone adaptation is not ingeneral a conserved, hyperbolic system. This stems from thephysiology where density does not flow through the domainlike a fluid leaving or entering only at the boundary, butinstead changes with sources (osteoblasts) and sinks (osteo-clasts) scattered throughout the domain. Alternatively, thefinite element method could be used but will in general betoo computationally intensive. This section is motivated byconsidering the generalized problem of Equation 20a:

φt = L(φ) (21)

Justification for these techniques can be found elsewhere [23].1) Spatial Gradients: An approximation to the operator L

is needed. This is a sum of the advection and mean curvatureterms.

L(φ) = −a+ bκ (22)= Ladvection(φ) + Lmean curvature(φ) (23)

The mean curvature term can simply be expanded and appro-priate finite difference subsituted into the equation:

Lmean curvature(φ) = bκ|∇φ| (24)

=

(φyy + φzz)φ2x

+ (φzz + φxx)φ2y

+ (φxx + φyy)φ2z

−2φxφyφxy − 2φzφxφzx − 2φyφzφyz(φ2x + φ2

y + φ2z

) (25)

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(a) parameterization (b) Parameter Distribution (c) Topology Change (d) Smooth Representation

Fig. 2. Disadvantages of using a parametric representation. 2a) Complex structures such as cancellous bone are difficult to parameterize. 2b) Parameterswill bunch during the flow. 2c) Changes in topology require explicit merging and splitting rules. 2d) Surface representations are implicitely smoothed withdifferentiable parametric representation.

The advection term is more difficult because central-differenceapproximations to the gradient operator cause oscillations.Instead, an upwind solver must be used:

Ladvection(φ) = −a|∇φ| (26)

= −a√φ2x + φ2

y + φ2z (27)

= −a+√

(p+)2 + (q−)2 + (r+)2 + (s−)2 + (t+)2 + (u−)2

−a−√

(p−)2 + (q+)2 + (r−)2 + (s+)2 + (t−)2 + (u+)2(28)

where x+ = max(x, 0), x− = min(x, 0), and p through u areone-sided differences:

p = D−x φ q = D+

x φr = D−

y φ s = D+y φ

t = D−z φ u = D+

z φ(29)

These first-order derivatives can be replaced with weightedessential non-oscillator schemes if higher order accuracy isneeded [78].

2) Time Stepping: Next, the solution must be time stepped.This is done using a forward Euler approximation to the timederivative:

φn+1 = φn + ∆tL(φn) (30)

This can be extended with the Runge-Kutta method if a moreaccurate solver is needed.

3) Courant-Friedrich-Lewy Condition: The method will beunstable if the Courant-Friedrich-Lewy (CFL) condition isnot met [79]. The CFL condition states that the “numericaldomain of dependence must include the physical domain ofdependence”. In essence, this means that the surface cannottravel further than a voxel in a single iteration. The time stepcan be selected by the following equation:

α = ∆t

(|a|

min(∆x,∆y,∆z)+

2|b|min(∆x2,∆y2,∆z2)

)(31)

where ∆x, ∆y, and ∆z are the voxels edge lengths. α mustbe selected less than 1 to satisfy the CFL condition and isselected to be 0.5 in this work.

4) Reinitialization: During evolution, the embedding candeviate from a signed distance transform. Reinitialization isthe process of returning the embedding to a signed distancetransform [19]. In this work, a method is used that guaranteesthat the embedding does not change sign during reinitalization,and thus keeps volume conserved [80]. This is achieved bysolving the following partial differential equation with thesame finite difference method:

φτ + S(φ0) (|∇φ| − 1) = 0 (32)

where τ is a time-like parameter and S(·) is a regularizedapproximation to the sign function:

S(φ) =φ√

φ2 + |∇φ|2h2(33)

where h = min(∆x,∆y,∆z). The reinitalization equationis performed after every iteration, which could be relaxed ifcomputation time was a concern.

B. Embedding

Attention is now placed on initialization the embedding, φ.This is a challenging task as the signed distance transformof binary images exhibit quantization, limiting numericalaccuracy of the flow and ability to measure curvatures [2].Instead, a previously developed method is used that instantiatesthe embedding directly from the density image, skippingbinarization [3]. The method is quickly summarized below.

The central idea is to construct an embedding ψ that doesnot satisfy the Eikonal condition but shares a zero crossingwith the desired embedding φ. This is achieved by subtractinga density threshold to shift the desired density level set to zero.Furthermore, noise reduction methods can also be applied.This leads to the following definition of the intermediateembedding:

ψ = T −Gσ ∗ ρ (34)

where T is a density threshold, Gσ is a Gaussian blur of sizeσ, and ψ is an embedding whose zero level set is the implicitsurface (Equation 18). Having the intermediate embedding,the closest point method [81], [82], [83] is used to establishthe narrowband distances, which are then marched to theremainder of the domain using the high-order fast sweepingmethod [84]. The obtained embedding φ satisfies the recoverycondition, Eikonal condition, is unique, and has an order ofaccuracy greater than unity [85], [3].

C. Density Component Estimation

Having the density and embedding image, the phase den-sities ρmarrow and ρbone can be estimated. The central idea isthat the density image can be constructed from the embeddingimage knowing the two phase densities and the embedding:

ρ = ρboneθ(−φ) + ρmarrow [1− θ(−φ)] (35)

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where θ is the Heaviside step function. We follow here themethod of Chan and Vese [86] to estimated ρmarrow and ρbonefrom the density image ρ an embedding φ:

ρbone =

∫Ωρ(x)θ(−φ(x))dV∫Ωθ(−φ(x))dV

(36)

ρmarrow =

∫Ωρ(x) [1− θ(−φ(x))] dV∫Ω

[1− θ(−φ(x))] dV(37)

It is assumed that the phase densities do not change duringadaptation. As such, volumetric bone mineral density can beestimated from the volume fraction of bone:

BV/TV =

∫Ωθ(−φ(x))dV∫

ΩdV

(38)

vBMD = ρboneBV/TV + ρmarrow [1− BV/TV] (39)

We remark here that while masking is not explicitly performedin this study, it can be achieved using simple surface editingoperators on signed distance fields [87], [88].

D. Morphometry

Beyond densities, morphometry can be performed directlyfrom the embedding using a previously developed tech-nique [62]. First, mean curvature can be computed using thedivergence of the surface normals:

H =1

2∇ ·(∇φ|∇φ|

)(40)

We remark that H and κ differ by a factor of the surfacedimensionality:

H =κ

2(41)

κ is the physicist’s mean curvature while H is the geometer’smean curvature. Next, Gaussian curvature can be computed onthe implicit contour [23]:

K = −

∣∣∣∣∣∣∣∣φxx φxy φxz φxφyx φyy φyz φyφzx φzy φzz φzφx φy φz 0

∣∣∣∣∣∣∣∣|∇φ|4

(42)

Using mean and Gaussian curvature with definition of thevolume and area elements, the volume (V ), area (A), surfaceaverage mean curvature (〈H〉), total Gaussian curvature (K),and Euler–Poincare characteristic (χ) can be measured usingpreviously developed methods [86], [80], [62], [3]:

V =

∫Ω

θ(−φ)dV (43)

A =

∫Ω

δ(−φ)|∇φ|dV (44)

〈H〉 =

∫ΩHδ(−φ)|∇φ|dV∫

ΩdV

(45)

K =

∫Ω

Kδ(−φ)|∇φ|dV (46)

χ =K

2π(47)

From these definitions, standard bone morphometric mea-sures can be derived. The structure model index (SMI) canbe computed from average mean curvature, volume, andarea [89], [90], trabecular bone pattern factor (TBPf) canbe computed from average mean curvature alone [91], [92],and connectivity density (Conn.D) can be computed from theEuler–Poincare characteristic and the total volume [93]:

SMI = 12〈H〉VA

(48)

TBPf = 2〈H〉 (49)

Conn.D =1− χ/2TV

(50)

The purpose of the factor-of-two is outlined in the Appendix.Importantly, since connected component filtering cannot beeasily implemented on the embedding and disconnected par-ticles will form during the flow, Odgaard’s constraints on theBetti numbers — that there is no marrow cavities (β2 = 0)and only one foreground particle (β0 = 1) — cannot beguaranteed [93]. The importance of this conclusion is outlinedin the discussion.

V. EXPERIMENT

Two experiments are performed to demonstrate curvature-based bone adaptation. Ten cubes of bovine trabecular bonewere previously sawed to 10 mm in edge length and imaged ata nominal resolution of 20 µm [94]. Embedding was performedwith a threshold of T = 400 mg HA/cc and Gaussian filterof standard deviation σ = 20 µm. This dataset was previouslyused to validate the embedding method [3].

Images were embedded and geometric flows simulated usingthe level set method. Two parameter sets are studied asdescribed later. 30 years were simulated and morphometry wasperformed every 3 years directly from the embedding. Mor-phometry included the bone surface to volume ratio (BS/BV,mm−1), volumetric bone mineral density (vBMD, mg HA/cc),connectivity density (Conn.D, mm−3), and structure modelindex (SMI, −). For rendering, volumes were reduced toa 2 mm edge length cube in order to visualize individualtrabeculae and the marching cubes algorithm [95] was usedto directly extract the surface at an isocontour of zero.

A. Assigned Flow

Model parameters are selected such that they representphysiologically plausible losses. Selecting a = −1 µm yr−1

and b = 100 µm2 yr−1 would erode a rod of thickness 100 µmroughly 2 µm yr−1. Given that trabecular bone has an averagethickness around 100− 250 µm, this is a reasonable loss overa human lifespan of 100 years. Of course, the process is non-linear, and more loss will be experienced.

B. Volume-Preserving Flow

Next, a volume-preserving flow is investigated. To ensurethe flow is volume-preserving, the parameters a, b are se-lected equal to the average mean curvature across the surface:

a

b= 〈κ〉 (51)

8

Subject 〈H〉 mm−1 κ mm−1 b µm2 yr−1 a µmyr−1

1 2.21 4.42 100 0.4422 -0.13 -0.26 100 -0.0263 1.27 2.53 100 0.2534 0.31 0.62 100 0.0625 0.37 0.73 100 0.0736 1.25 2.51 100 0.2517 0.56 1.12 100 0.1128 2.11 4.22 100 0.4229 1.35 2.70 100 0.270

10 -0.02 -0.04 100 -0.004TABLE I

PER-SUBJECT PARAMETERS FOR CURVATURE DRIVEN BONE ADAPTATIONTHAT PRODUCE A VOLUME-PRESERVING FLOW.

The same curvature constant, b = 100 µm2 yr−1, is used butthe propagation constant, a, is allowed to vary with eachsubject. These values are given in Table I. The volume-preserving flow is interesting because it implies changes inthe bone surface that fundamentally cannot be measured byareal or volumetric bone volume fraction, requiring imagingof the microarchitecture. Furthermore, this suggests a deeperrelationship between structure and calcium homeostasis wherebone can be turning over but net calcium flux through digestionand excretion is zero.

VI. RESULTS

A. Assigned Flow

Morphometry during curvature-based bone adaptation isplotted in Figure 3. Bone surface to volume ratio increasesnon-linearly with time. Similarly, structure model index in-creases with time. Owing to the inverse relationship betweenSMI and BS/BV, the changes in SMI must be driven by anincrease in mean curvature across the surface. Bone mineraldensity decreases almost linearly and nearly at the same ratefor all subjects. A subject specific response was seen inconnectivity density. While a few subjects increased Conn.Dacross the time frame, others increased then decreased rapidly.The number of connections can rise when plates form holes,decrease when rods disconnect, and effectively decrease bythe formation of isolated particles, making this morphometricoutcome difficult to interpret. The surface of the mediansubject by density is visualized in Figure 4. Rods are seendisconnecting and resorbing, the structure thins throughoutand becomes more porous. Furthermore, the surface loosesroughness in the first 3 years consistent with noise beingremoved by mean curvature flow.

B. Volume-Preserving Flow

Morphometrics for the volume-preserving flow are plottedin Figure 5. As expected, bone mineral density decreasesonly slightly, a result seen in a previous study [80]. Theratio of surface area to volume decreases over time whilestructure model index increases. This is driven by the inverserelationship between SMI and BS/BV where surface averagemean curvature is constant in the volume-preserving flow.Since volume is constant, the surface area must be decreasing,consistent with area being the first variation of volume. Finally,connectivity density increases for approximately 10 years then

starts to decrease. This is consistent with particles forming,increasing the Euler–Poincare characteristic, followed by rodsdisconnecting. The initial, 15 year, and 30 year epochs of themedian density subject are rendered in Figure 6. Structuralchanges are subtle, but thin rods can be observed disconnectingand negative curvature areas thickening. As in the non-volume-preserving case, noise on the surface is rapidly removed.

VII. DISCUSSION

Bone adaptation is presented as a geometric flow. Curvature-based bone adaptation is presented as the continuous versionof the discrete simulated bone atrophy method. The geometricflow can be simulated using the level set method, whichnaturally handles topological changes. Two parameter setswere investigated, one resembling age-related bone loss andanother being a volume-preserving flow.

The concretized model of curvature-based bone adaptation,and its predecessor simulated bone atrophy [50], are unlikelyto accurately predict in vivo bone microarchitectural changes.This owes to the relationship of the models to the Young-Laplace equation of surface tension, giving bubble-like ar-chitecutres if the model runs longer than 30 years. However,the power of these models is in their clarity and computationalabilities. As with simulated bone atrophy, topological changescan be handled naturally. Furthermore, the methods of sim-ulated bone atrophy were central to developing a load-drivenmodel that could handle topological changes [46]. In this way,the specific instantiation (curvature-based bone adaptation) andgeneral theory (bone adaptation as a geometric flow) shouldbe separated [96].

Beyond a specific instantiation, there are limitations todescribing bone adaptation as a geometric flow. The centralassumption to treat bone as a geometric flow is that thebone is orientable and smooth. This assumption cannot bemet during fetal development and fracturing healing wheremineralization processes are not occurring on an existingsurface. Generally, ontogenesis and fracturing healing will bewell described by internal remodeling methods [25] whilemodeling and remodeling will be well described by externalremodeling methods. Such a situation alludes to a descriptionthat is not internal nor external remodeling. There should bean underlying process which gives rise to internal or externalremodeling based on circumstance.

In a search for the underlying dynamics of bone biology,a natural pattern similar to trabecular bone was sought. Anastonishing similarity is seen between trabecular bone and Tur-ing patterns [97]. Turing patterns emerge from simple reaction-diffusion equations, giving wonderfully complex shapes. TheGray-Scott model is arguably the most studied of these mod-els [98] and extensive work has been done to classify thepatterns as a function of their parameters [99]. Another famousreaction-diffusion model is the Allen-Cahn equation [100],which was shown to converge to mean curvature flow [101].Understanding trabecular patterning as a consequence of thedynamics of biochemistry will provide a deeper understandingof the multi-scale link in bone biology [102].

The obvious next step is to incorporate the level set methodinto a functional adaptation model. Developing a functional

9

5

10

15

20

25

30

35

0 10 20 30

Time [years]

BS

/BV

[1/m

m]

(a) BS/BV

0

100

200

300

0 10 20 30

Time [years]

vBM

D [m

g H

A/c

cm]

(b) vBMD

0

5

10

15

0 10 20 30

Time [years]

Con

n.D

[1/m

m3 ]

(c) Conn.D

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30

Time [years]

SM

I [−

]

(d) SMI

Fig. 3. Measured morphometry during curvature based bone adaptation with parametes a = −1 µmyr−1 and b = 100 µm2 yr−1. Bone (3a) surface areato volume ratio, (3b) volumetric minearl density, (3c) connectivity density, and (3d) structure model index are plotted every three years over 30 years ofsimulation. Connected lines are individual subjects.

adaptation model using the signed distance embedding is aconcatenation of the biphasic model [3] and existing load-driven models [29], [28], [103]. The key distinction is to per-form finite element analysis on the density image constructedfrom the biphasic solution while having motion of the bonesurface on the embedding. This has the major advantage of nothaving to perform connected components filtering for finiteelement analysis, where isolated parts of bone relevant formeasuring changes in total calcium must be removed in orderto permit a solution of the finite element model.

Without binarizing the volume, it is difficult to perform con-nected components filtering during embedding or adaptationwhere isolated bone tissue components can arise. Connectedcomponents filtering is needed to meet the assumptions ofOdgaard in measuring connectivity density [93]. The methodpresented here cannot assess the Betti numbers directly, butconnectivity density is still estimated so the numerics are inter-pretable. However, alternative methods exist for computing theBetti numbers of an implicit curve [104], which would allowthe direct measurement of β1 without connected componentsfiltering. Furthermore, this method may be less sensitive thanthe total Gaussian curvature method used here. Together withperforming finite element analysis on the constructed density

image, this would permit the development of disconnectedcomponents during adaptation, which would be important formonitoring calcium homeostasis. Odgaard identified the issueof isolated bone particles during adaptation in his connectivitydensity work [93]:

During bone formation and bone healing isolatedislands of bone may exist, but these and relatedexceptions will not be considered further. The mainreason for neglecting these exceptions is that fullyisolated bony strands do only contribute very little, ifat all, to the mechanical competence of a cancellousbone region.

A major question that remains to be resolved is how toincorporate a local strain field and a local advection forcetogether. Existing models assume the magnitude of the dif-feomorphism caused by the strain field is much smaller inmagnitude than local changes in bone morphometry, consistentwith the small-strain assumption of mechanics. It is not obvi-ous how to validate this assumption, nor how to incorporatethe two types of motion together. However, incorporating thestrain field with the level set equation would provide a clearpath to model dynamic behavior [105], a currently under-modeled aspect of bone adaptation.

10

(a) t = 0yr (b) t = 3yr (c) t = 6yr (d) t = 9yr

(e) t = 12 yr (f) t = 15 yr (g) t = 18 yr (h) t = 21 yr

(i) t = 24 yr (j) t = 27 yr (k) t = 30 yr

Fig. 4. Visualization of the bone surface changing across the simulation timeframe. Rods disconnect exhibiting the ability of level set methods to capturetopological changes. The medial subject by bone mineral density is displayed.

Lastly, curvature-based bone adaptation is intricately linkedto surfaces of constant mean curvature. These theories presenta relationship between a functional being minimized (a La-grangian) and a surface minimizing that Lagrangian, sug-gesting that bone can be a minimal surface of a measuredifferent from curvature. The concept of bone being optimalin some sense dates back to Wolff, with early formalizationof the problem using Lagrangians and optimal control theorydating to Carter [106]. Early work on topology optimizationgiven prescribed mechanical competence demonstrates struc-tures remarkably similar to the cross-section of the humeralhead [107]. Deriving Frost’s mechanostat — the adaptationfunction, F — as the Euler-Lagrange of a Lagrangian mayfinally answer Huiskes [71]: If bone is the answer, then whatis the question?

APPENDIXCORRECTIONS TO ODGAARD

Error in Odgaard’s work on connectivity density [93] isstraightforward to assess. It arises from the difference indimensionality of the surface, where Odgaard’s work didnot modify the equations for Euler–Poincare characteristicbetween two-dimensional and three-dimensional structures.The error can be seen immediately in Odgaard’s writing wherehe states [93]:

For any solid body which can be deformed into asolid sphere, the Euler characteristic is 1; for any

body which can be deformed into a solid torus, theEuler characteristic is 0, and generally

χ(v) = 1− n

for a solid sphere with n handles.This is true in two dimensions but the equation

χ = 2− 2g (52)

is appropriate in three dimensions, with g the surface genus.This oversight permeates the computation. The correction thenis to multiply the calculated Euler–Poincare characteristic bytwo, which corresponds roughly to doubling the connectivitydensity for large values of χ. Similarly, considering the objectis solid, β2 should be set to 1, but this is a minor correctiongiven the highly negative Euler–Poincare characteristic oftrabecular bone.

One can easily assess their morphometry softwarefor this flaw by performing connectivity density on athree-dimensional sphere and torus, observing the Euler–Poincare characteristic. The ideal results should be 2 for thesphere and 0 for the torus. Software implemented for twodimensions will return 1 and 0. A double-handled torus wouldreturn an Euler–Poincare characteristic of -1 when it shouldreturn -2.

There is no obvious way to correct this mistake in theliterature. Many papers have measured bone as having halfits true connectivity density. A one-time backwards incompat-ible change to the field would cause confusion. The authors

11

5

10

15

20

25

0 10 20 30

Time [years]

BS

/BV

[1/m

m]

(a) BS/BV

0

100

200

300

0 10 20 30

Time [years]

vBM

D [m

g H

A/c

cm]

(b) vBMD

0.0

2.5

5.0

7.5

10.0

0 10 20 30

Time [years]

Con

n.D

[1/m

m3 ]

(c) Conn.D

−0.5

0.0

0.5

1.0

1.5

2.0

0 10 20 30

Time [years]

SM

I [−

]

(d) SMI

Fig. 5. Measured morphometry during the volume-preserving flow. Bone (3a) surface area to volume ratio, (3b) volumetric minearl density, (3c) connectivitydensity, and (3d) structure model index are plotted every three years over 30 years of simulation. Volumetric bone mineral density does not change as expectedfrom the model. Connected lines are individual subjects.

recommend the field continues as-is with a note in the marginsabout bone actually being twice as connected as measured.

REFERENCES

[1] B. A. Besler, L. Gabel, L. A. Burt, N. D. Forkert, and S. K. Boyd,“Bone adaptation as level set motion,” in International Workshop onComputational Methods and Clinical Applications in MusculoskeletalImaging. Springer, 2018, pp. 58–72.

[2] B. A. Besler, T. D. Kemp, N. D. Forkert, and S. K. Boyd, “Artifacts ofquantization in distance transforms,” arXiv preprint arXiv:2011.08880,2020.

[3] ——, “Constructing high-order signed distance maps from com-puted tomography data with application to bone morphometry,” arXivpreprint arXiv:2111.01350, 2021.

[4] H. M. Frost, “Tetracycline-based histological analysis of bone remod-eling,” Calcified tissue research, vol. 3, no. 1, pp. 211–237, 1969.

[5] ——, “Bone “mass” and the “mechanostat”: a proposal,” The anatom-ical record, vol. 219, no. 1, pp. 1–9, 1987.

[6] ——, “The utah paradigm of skeletal physiology: an overview of itsinsights for bone, cartilage and collagenous tissue organs,” Journal ofbone and mineral metabolism, vol. 18, no. 6, pp. 305–316, 2000.

[7] A. D. Berendsen and B. R. Olsen, “Bone development,” Bone, vol. 80,pp. 14–18, 2015.

[8] R. Marsell and T. A. Einhorn, “The biology of fracture healing,” Injury,vol. 42, no. 6, pp. 551–555, 2011.

[9] L. J. Raggatt and N. C. Partridge, “Cellular and molecular mechanismsof bone remodeling,” Journal of Biological Chemistry, vol. 285, no. 33,pp. 25 103–25 108, 2010.

[10] E. F. Eriksen, “Cellular mechanisms of bone remodeling,” Reviews inEndocrine and Metabolic Disorders, vol. 11, no. 4, pp. 219–227, 2010.

[11] G. Beaupre, T. Orr, and D. Carter, “An approach for time-dependentbone modeling and remodeling—theoretical development,” Journal ofOrthopaedic Research, vol. 8, no. 5, pp. 651–661, 1990.

[12] E. Kreyszig, “Differential geometry,” in Differential Geometry. Uni-versity of Toronto Press, 2019.

[13] B. B. Mandelbrot, The fractal geometry of nature. WH freeman NewYork, 1982, vol. 1.

[14] S. Majumdar, R. S. Weinstein, and R. R. Prasad, “Application of fractalgeometry techniques to the study of trabecular bone,” Medical physics,vol. 20, no. 6, pp. 1611–1619, 1993.

[15] N. Fazzalari and I. Parkinson, “Fractal dimension and architecture oftrabecular bone,” The Journal of pathology, vol. 178, no. 1, pp. 100–105, 1996.

[16] W. Geraets and P. Van Der Stelt, “Fractal properties of bone.” Den-tomaxillofacial Radiology, vol. 29, no. 3, pp. 144–153, 2000.

[17] A. Boyde, E. Maconnachie, S. Reid, G. Delling, and G. Mundy,“Scanning electron microscopy in bone pathology: review of methods,potential and applications,” Scanning electron microscopy, vol. 1986,no. 4, p. 31, 1986.

[18] B. L. Jorgenson, H. R. Buie, D. D. McErlain, C. Sandino, and S. K.Boyd, “A comparison of methods for in vivo assessment of corticalporosity in the human appendicular skeleton,” Bone, vol. 73, pp. 167–175, 2015.

[19] M. Sussman, P. Smereka, and S. Osher, “A level set approach forcomputing solutions to incompressible two-phase flow,” Journal ofComputational physics, vol. 114, no. 1, pp. 146–159, 1994.

[20] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contourmodels,” International journal of computer vision, vol. 1, no. 4, pp.321–331, 1988.

[21] G. Huisken et al., “Flow by mean curvature of convex surfaces into

12

(a) Initial (b) 15 Years (c) 30 Years

Fig. 6. Visualization of the surface during volume-preserving flow at three epochs. Rods can be seen disconnecting, suggesting a decrease in mechanicalcompetence.

spheres,” Journal of Differential Geometry, vol. 20, no. 1, pp. 237–266,1984.

[22] M. Gage and R. S. Hamilton, “The heat equation shrinking convexplane curves,” Journal of Differential Geometry, vol. 23, no. 1, pp.69–96, 1986.

[23] J. A. Sethian, Level set methods and fast marching methods: evolvinginterfaces in computational geometry, fluid mechanics, computer vision,and materials science. Cambridge university press, 1999, vol. 3.

[24] M. G. Crandall and P.-L. Lions, “Viscosity solutions of hamilton-jacobiequations,” Transactions of the American mathematical society, vol.277, no. 1, pp. 1–42, 1983.

[25] D. R. Carter, “Mechanical loading histories and cortical bone remodel-ing,” Calcified tissue international, vol. 36, no. 1, pp. S19–S24, 1984.

[26] H. M. Frost, “From wolff’s law to the utah paradigm: insights aboutbone physiology and its clinical applications,” The Anatomical Record:An Official Publication of the American Association of Anatomists, vol.262, no. 4, pp. 398–419, 2001.

[27] T. Adachi, Y. Tomita, H. Sakaue, and M. Tanaka, “Simulation oftrabecular surface remodeling based on local stress nonuniformity,”JSME Int J, vol. 40, no. 4, pp. 782–792, 1997.

[28] T. Adachi, K. Tusbota, Y. Tomita, and S. Hollister, “Trabecular surfaceremodeling simulation for cancellous bone using microstructural voxelfinite element models,” J Biomech Eng, vol. 123, pp. 403–409, 2001.

[29] R. Huiskes, R. Ruimerman, G. van Lenthe, and J. Janssen, “Effects ofmechanical forces on maintenance and adaptation of form in trabecularbone,” Nature, vol. 405, pp. 704–406, 2000.

[30] F. A. Schulte, F. M. Lambers, D. J. Webster, G. Kuhn, and R. Muller,“In vivo validation of a computational bone adaptation model usingopen-loop control and time-lapsed micro-computed tomography,” Bone,vol. 49, pp. 1166–1172, 2011.

[31] Y. Kameo and T. Adachi, “Modeling trabecular bone adaptation tolocal bending load regulated by mechanosensing osteocytes,” ActaMechanica, vol. 225, no. 10, pp. 2833–2840, 2014.

[32] P. Christen, K. Ito, R. Ellouz, S. Boutroy, E. Sornay-Rendu, R. D.Chapurlat, and B. Van Rietbergen, “Bone remodelling in humans isload-driven but not lazy,” Nature communications, vol. 5, no. 1, pp.1–5, 2014.

[33] S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims, and L. M. Wahl,“Mathematical model predicts a critical role for osteoclast autocrineregulation in the control of bone remodeling,” Bone, vol. 33, no. 2, pp.206–215, 2003.

[34] V. Lemaire, F. L. Tobin, L. D. Greller, C. R. Cho, and L. J. Suva,“Modeling the interactions between osteoblast and osteoclast activitiesin bone remodeling,” Journal of theoretical biology, vol. 229, no. 3,pp. 293–309, 2004.

[35] P. Pivonka and S. V. Komarova, “Mathematical modeling in bone biol-ogy: From intracellular signaling to tissue mechanics,” Bone, vol. 47,no. 2, pp. 181–189, 2010.

[36] F. A. Gerhard, D. J. Webster, G. H. Van Lenthe, and R. Muller,“In silico biology of bone modelling and remodelling: adaptation,”Philosophical Transactions of the Royal Society A: Mathematical,

Physical and Engineering Sciences, vol. 367, no. 1895, pp. 2011–2030,2009.

[37] D. Webster and R. Muller, “In silico models of bone remodeling frommacro to nano—from organ to cell,” Wiley Interdisciplinary Reviews:Systems Biology and Medicine, vol. 3, no. 2, pp. 241–251, 2011.

[38] J. Von Neumann, “The general and logical theory of automata,” 1951,pp. 1–41, 1951.

[39] C. G. Langton, “Studying artificial life with cellular automata,” PhysicaD: Nonlinear Phenomena, vol. 22, no. 1-3, pp. 120–149, 1986.

[40] S. Wolfram, A new kind of science. Wolfram media Champaign, IL,2002, vol. 5.

[41] P. R. Buenzli, J. Jeon, P. Pivonka, D. W. Smith, and P. T. Cummings,“Investigation of bone resorption within a cortical basic multicellularunit using a lattice-based computational model,” Bone, vol. 50, no. 1,pp. 378–389, 2012.

[42] A. Tovar, “Bone remodeling as a hybrid cellular automaton optimiza-tion process,” Thesis, 2004.

[43] D. Tourolle, “A micro-scale multiphysics framework for fracture heal-ing and bone remodelling,” Ph.D. dissertation, ETH Zurich, 2019.

[44] G. Beaupre, T. Orr, and D. Carter, “An approach for time-dependentbone modeling and remodeling—application: A preliminary remodel-ing simulation,” Journal of Orthopaedic Research, vol. 8, no. 5, pp.662–670, 1990.

[45] S. C. Cowin, R. T. Hart, J. R. Balser, and D. H. Kohn, “Functionaladaptation in long bones: establishing in vivo values for surfaceremodeling rate coefficients,” J Biomech, vol. 18, no. 9, pp. 665–84,1985.

[46] F. A. Schulte, A. Zwahlen, F. M. Lambers, G. Kuhn, D. Ruffoni,D. Betts, D. J. Webster, and R. Muller, “Strain-adaptive in silicomodeling of bone adaptation—a computer simulation validated by invivo micro-computed tomography data,” Bone, vol. 52, no. 1, pp. 485–492, 2013.

[47] R. Hart, D. Davy, and K. Heiple, “Mathematical modeling and numer-ical solutions for functionally dependent bone remodeling,” CalcifiedTissue International, vol. 36, pp. S104–S109, 1984.

[48] H. Weinans, R. Huiskes, and H. Grootenboer, “The behavior of adaptivebone-remodeling simulation models,” Journal of biomechanics, vol. 25,no. 12, pp. 1425–1441, 1992.

[49] S. C. Cowin and K. Firoozbakhsh, “Bone remodeling of diaphysialsurfaces under constant load: theoretical predictions,” Journal ofBiomechanics, vol. 14, no. 7, pp. 471–484, 1981.

[50] R. Muller and P. Ruegsegger, “Analysis of mechanical propertiesof cancellous bone under conditions of simulated bone atrophy,” JBiomech, vol. 29, no. 8, pp. 1053–60, 1996.

[51] R. Mueller and W. C. Hayes, “Biomechanical competence of mi-crostructural bone in the progress of adaptive bone remodeling,” inDevelopments in X-ray tomography, vol. 3149. International Societyfor Optics and Photonics, 1997, pp. 69–81.

[52] W. Pistoia, B. Van Rietbergen, and P. Ruegsegger, “Mechanical conse-quences of different scenarios for simulated bone atrophy and recoveryin the distal radius,” Bone, vol. 33, no. 6, pp. 937–945, 2003.

13

[53] R. Muller, “Long-term prediction of three-dimensional bone architec-ture in simulations of pre-, peri-and post-menopausal microstructuralbone remodeling,” Osteoporosis International, vol. 16, no. 2, pp. S25–S35, 2005.

[54] P. Laplace, Traite de Mecanique Celeste: Tome IV. A Paris: Chez J.B. M. Duprat, 1805, vol. 4.

[55] T. Young, “Iii. an essay on the cohesion of fluids,” Philosophicaltransactions of the royal society of London, no. 95, pp. 65–87, 1805.

[56] B. A. Besler, T. D. Kemp, N. D. Forkert, and S. K. Boyd, “Sphere underadvection and mean curvature flow,” arXiv preprint arXiv:2012.10483,2020.

[57] A. H. Schoen, Infinite periodic minimal surfaces without self-intersections. National Aeronautics and Space Administration, 1970.

[58] D. L. Chopp, “Computing minimal surfaces via level set curvatureflow,” 1991.

[59] D. L. Chopp and J. A. Sethian, “Flow under curvature: singularity for-mation, minimal surfaces, and geodesics,” Experimental Mathematics,vol. 2, no. 4, pp. 235–255, 1993.

[60] S. C. Kapfer, S. T. Hyde, K. Mecke, C. H. Arns, and G. E. Schroder-Turk, “Minimal surface scaffold designs for tissue engineering,” Bio-materials, vol. 32, no. 29, pp. 6875–6882, 2011.

[61] L. Zhang, S. Feih, S. Daynes, S. Chang, M. Y. Wang, J. Wei, andW. F. Lu, “Energy absorption characteristics of metallic triply periodicminimal surface sheet structures under compressive loading,” AdditiveManufacturing, vol. 23, pp. 505–515, 2018.

[62] B. A. Besler, T. D. Kemp, A. S. Michalski, N. D. Forkert, and S. K.Boyd, “Local morphometry of closed, implicit surfaces,” arXiv preprintarXiv:2108.04354, 2021.

[63] A. Witkin, “Scale-space filtering: A new approach to multi-scale de-scription,” in ICASSP’84. IEEE International Conference on Acoustics,Speech, and Signal Processing, vol. 9. IEEE, 1984, pp. 150–153.

[64] J. J. Koenderink, “The structure of images,” Biological cybernetics,vol. 50, no. 5, pp. 363–370, 1984.

[65] B. Merriman, J. K. Bence, and S. Osher, Diffusion generated motion bymean curvature. Department of Mathematics, University of California,Los Angeles, 1992.

[66] L. C. Evans, “Convergence of an algorithm for mean curvature motion,”Indiana University mathematics journal, pp. 533–557, 1993.

[67] J. F. Blinn, “Jim blinn’s corner-return of the jaggy (high frequencyfiltering),” IEEE Computer Graphics and Applications, vol. 9, no. 2,pp. 82–89, 1989.

[68] A. M. Parfitt, M. K. Drezner, F. H. Glorieux, J. A. Kanis, H. Malluche,P. J. Meunier, S. M. Ott, and R. R. Recker, “Bone histomorphometry:standardization of nomenclature, symbols, and units: report of theasbmr histomorphometry nomenclature committee,” Journal of boneand mineral research, vol. 2, no. 6, pp. 595–610, 1987.

[69] F. A. Schulte, F. M. Lambers, G. Kuhn, and R. Muller, “In vivo micro-computed tomography allows direct three-dimensional quantification ofboth bone formation and bone resorption parameters using time-lapsedimaging,” Bone, vol. 48, no. 3, pp. 433–442, 2011.

[70] G. Sapiro, Geometric partial differential equations and image analysis.Cambridge university press, 2006.

[71] R. Huiskes, “If bone is the answer, then what is the question?” Journalof anatomy, vol. 197, no. 2, pp. 145–156, 2000.

[72] J. Ashburner and K. J. Friston, “Voxel-based morphometry—the meth-ods,” Neuroimage, vol. 11, no. 6, pp. 805–821, 2000.

[73] F. L. Bookstein, “Landmark methods for forms without landmarks:morphometrics of group differences in outline shape,” Medical imageanalysis, vol. 1, no. 3, pp. 225–243, 1997.

[74] Y. Pauchard, C. Mattmann, A. Kuhn, J. A. Gasser, and S. K. Boyd,“European society of biomechanics sm perren award 2008: Usingtemporal trends of 3d bone micro-architecture to predict bone quality,”Journal of biomechanics, vol. 41, no. 14, pp. 2946–2953, 2008.

[75] A. Dervieux and F. Thomasset, “A finite element method for thesimulation of a rayleigh-taylor instability,” ser. Approximation Methodsfor Navier-Stokes Problems. Springer Berlin Heidelberg, 1980,Conference Proceedings, pp. 145–158.

[76] ——, “Multifluid incompressible flows by a finite element method,”ser. Seventh International Conference on Numerical Methods in FluidDynamics. Springer Berlin Heidelberg, 1981, Conference Proceedings,pp. 158–163.

[77] S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations,”Journal of computational physics, vol. 79, no. 1, pp. 12–49, 1988.

[78] X.-D. Liu, S. Osher, and T. Chan, “Weighted essentially non-oscillatoryschemes,” Journal of computational physics, vol. 115, no. 1, pp. 200–212, 1994.

[79] R. Courant, K. Friedrichs, and H. Lewy, “Uber die partiellen differen-zengleichungen der mathematischen physik,” Mathematische annalen,vol. 100, no. 1, pp. 32–74, 1928.

[80] D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, “A pde-basedfast local level set method,” Journal of computational physics, vol. 155,no. 2, pp. 410–438, 1999.

[81] D. L. Chopp, “Some improvements of the fast marching method,” SIAMJournal on Scientific Computing, vol. 23, no. 1, pp. 230–244, 2001.

[82] S. J. Ruuth and B. Merriman, “A simple embedding method for solvingpartial differential equations on surfaces,” Journal of ComputationalPhysics, vol. 227, no. 3, pp. 1943–1961, 2008.

[83] M. Coquerelle and S. Glockner, “A fourth-order accurate curvaturecomputation in a level set framework for two-phase flows subjected tosurface tension forces,” Journal of Computational Physics, vol. 305,pp. 838–876, 2016.

[84] Y.-T. Zhang, H.-K. Zhao, and J. Qian, “High order fast sweepingmethods for static hamilton–jacobi equations,” Journal of ScientificComputing, vol. 29, no. 1, pp. 25–56, 2006.

[85] B. A. Besler, T. D. Kemp, N. D. Forkert, and S. K. Boyd, “High-order signed distance transform of sampled signals,” arXiv preprintarXiv:2110.13354, 2021.

[86] T. F. Chan and L. A. Vese, “Active contours without edges,” IEEETransactions on image processing, vol. 10, no. 2, pp. 266–277, 2001.

[87] H.-K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit andnonparametric shape reconstruction from unorganized data using a vari-ational level set method,” Computer Vision and Image Understanding,vol. 80, no. 3, pp. 295–314, 2000.

[88] K. Museth, D. E. Breen, R. T. Whitaker, and A. H. Barr, “Level setsurface editing operators,” in Proceedings of the 29th annual conferenceon Computer graphics and interactive techniques, 2002, pp. 330–338.

[89] T. Hildebrand and P. Ruegsegger, “Quantification of bone microar-chitecture with the structure model index,” Computer Methods inBiomechanics and Bio Medical Engineering, vol. 1, no. 1, pp. 15–23,1997.

[90] H. Jinnai, H. Watashiba, T. Kajihara, Y. Nishikawa, M. Takahashi,and M. Ito, “Surface curvatures of trabecular bone microarchitecture,”Bone, vol. 30, no. 1, pp. 191–194, 2002.

[91] M. Hahn, M. Vogel, M. Pompesius-Kempa, and G. Delling, “Trabecularbone pattern factor—a new parameter for simple quantification of bonemicroarchitecture,” Bone, vol. 13, no. 4, pp. 327–330, 1992.

[92] M. Stauber and R. Muller, “Volumetric spatial decomposition oftrabecular bone into rods and plates—a new method for local bonemorphometry,” Bone, vol. 38, no. 4, pp. 475–484, 2006.

[93] A. Odgaard and H. Gundersen, “Quantification of connectivity incancellous bone, with special emphasis on 3-d reconstructions,” Bone,vol. 14, no. 2, pp. 173–182, 1993.

[94] C. Sandino and S. K. Boyd, “Trabecular bone poroelasticityfor microct-based fe models,” in Computational Biomechanics forMedicine. Springer, 2013, pp. 145–155.

[95] W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution3d surface construction algorithm,” ACM siggraph computer graphics,vol. 21, no. 4, pp. 163–169, 1987.

[96] P. A. M. Dirac, “The evolution of the physicist’s picture of nature,”Scientific American, vol. 208, no. 5, pp. 45–53, 1963.

[97] A. M. Turing, “The chemical basis of morphogenesis,” PhilosophicalTransactions of the Royal Society of London. Series B, BiologicalSciences, vol. 237, no. 641, pp. 37–72, 1952.

[98] P. Gray and S. Scott, “Autocatalytic reactions in the isothermal,continuous stirred tank reactor: Oscillations and instabilities in thesystem a+ 2b→ 3b; b→ c,” Chemical Engineering Science, vol. 39,no. 6, pp. 1087–1097, 1984.

[99] J. E. Pearson, “Complex patterns in a simple system,” Science, vol.261, no. 5118, pp. 189–192, 1993.

[100] S. M. Allen and J. W. Cahn, “A microscopic theory for antiphaseboundary motion and its application to antiphase domain coarsening,”Acta metallurgica, vol. 27, no. 6, pp. 1085–1095, 1979.

[101] T. Ilmanen et al., “Convergence of the allen-cahn equation to brakke’smotion by mean curvature,” J. Differential Geom, vol. 38, no. 2, pp.417–461, 1993.

[102] D. Webster and R. Muller, “In silico models of bone remodeling frommacro to nano—from organ to cell,” Wiley Interdisciplinary Reviews:Systems Biology and Medicine, vol. 3, no. 2, pp. 241–251, 2011.

[103] F. A. Schulte, D. Ruffoni, F. M. Lambers, D. Christen, D. J. Webster,G. Kuhn, and R. Muller, “Local mechanical stimuli regulate boneformation and resorption in mice at the tissue level,” PloS one, vol. 8,no. 4, p. e62172, 2013.

14

[104] V. Pascucci and K. Cole-McLaughlin, “Efficient computation of thetopology of level sets,” in IEEE Visualization, 2002. VIS 2002. IEEE,2002, pp. 187–194.

[105] C. H. Turner, “Three rules for bone adaptation to mechanical stimuli,”Bone, vol. 23, no. 5, pp. 399–407, 1998.

[106] C. R. Jacobs, J. C. Simo, G. S. Beaupre, and D. R. Carter, “Adaptive

bone remodeling incorporating simultaneous density and anisotropyconsiderations,” Journal of biomechanics, vol. 30, no. 6, pp. 603–613,1997.

[107] J. A. Sethian and A. Wiegmann, “Structural boundary design via levelset and immersed interface methods,” Journal of computational physics,vol. 163, no. 2, pp. 489–528, 2000.